The literature of curve interpolation techniques, otherwise known as “splines” is vast. A system for interactive design of curves should have the following properties: (a) The system should be capable of accurately representing a rich variety of curves with a minimum of input (measured as the number of control points); (b) The resulting curves should be smooth; (c) When moving control points, the effect on the curve should be direct and simple to understand; and (d) The computational resources should be modest enough to allow for update of the display at interactive rates.
Much curve design, including most font design, is done using cubic Bezier curves. FIG. 3 shows a typical cubic Bezier segment with knots and additional control points. In a Bezier curve, some of the control points (“knots or knot points”) lie on the curve (30, 36), but others lie off the curve (32, 34), and the position of the off-curve control points controls the shape of the curve. As the designer moves control points 32 and 34, the shape of the curve between control points 30 and 36 changes. This technique is simple to compute and very fast, and is good at representing a rich diversity of curves, but is not very simple to control. It is especially difficult to achieve smooth curves through the on-curve control points, and it takes considerable time and effort for designers to learn to do so.
A particularly difficult aspect of curve design for font generation is making smooth transitions between straight and curved sections. Smoothness is a subjective quality, and there is no consensus in the literature on the best way to measure it, but one very commonly related objective measure closely connected to smoothness is the degree of continuity across the knot points. This continuity measure, in turn, is defined in terms of “curvature”. Curvature is defined as the inverse of instantaneous radius, i.e., the inverse of the radius of the osculating circle that “kisses” the curve at every point. If the curvature exists at all points (i.e. there are no sharp corners), then the curve has G1 continuity.
If the curvature is also continuous through the control point, then it has G2 continuity. Higher degrees of continuity are defined in terms of derivatives of the curvature with respect to arclength. If the first derivative of curvature is continuous, then the curve has G3 continuity, and if the second derivative of curvature is also continuous, then the curve has G4 continuity. In general, cubic Bezier curves only guarantee G1 continuity.
Another class of curve generation techniques involves “interpolating splines,” or curves that go through (“interpolate”) all the control points. One example used in practice for font design is “Smooth, Easy to Compute Interpolating Splines,” by John Hobby, Stanford University, 1985 as implemented in the METAFONT system. In the prior art METAFONT spline, the range of choices for constraints is limited (the constraint is that of continuity of “mock curvature” across control points). In curve design, a primitive is defined as the basic element upon which more complex curves are built. The underlying primitive curve in the METAFONT system is a reduced family of cubic polynomials (with a two-dimensional parameter space). Another prior art system known as Ikarus, by Peter Karow, is similar to METAFONT.
Yet another class of interpolating spline techniques is the “variational approach,” in which a curve is chosen to minimize some energy functional. Such techniques emulate mechanical splines based on thin strips of elastic material such as metal or wood. Examples include Gumhold (Designing Optimal Curves in 2D, Proc. CEIG, July 2004 pages 61-76), and Moreton (Minimum Curvature Variation Curves, Networks and Surfaces for Fair Free-Form Shape Design, 1992 University of California, Berkeley Calif.). In general, variational techniques provide for a high degree of smoothness, but also require substantial computational resources. Moreton described MEC (Minimum Energy Curve) with G2 continuity and also MVC (Minimum Variation of Curvature) splines with G4 continuity.